A triangle is a closed figure formed by three intersecting line segments. Triangles are classified based on sides and angles:
By Side Lengths:
Equilateral Triangle: All three sides are equal; all three angles are equal to 60°.
Isosceles Triangle: Exactly two sides are equal; the angles opposite to the equal sides are equal.
Scalene Triangle: All three sides and all three angles have distinct measures.
By Angle Measures:
Acute-angled: All three interior angles are strictly less than 90°.
Right-angled: Exactly one interior angle is 90°.
Obtuse-angled: Exactly one interior angle is strictly greater than 90°.
B. Fundamental Theorems & Properties
Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be strictly greater than the third side:
a + b > c, b + c > a, a + c > b
Area Comparison (Equilateral Triangle vs. Square):
For a shared side length s:
Area(Square) = s² | Area(Equilateral Triangle) = (√3 / 4) × s² ≈ 0.433 s²
Since 1 > 0.433, a square always encloses significantly more area than an equilateral triangle of identical side length.
Similarity Criteria: Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional:
AA (Angle-Angle) Criterion: If two pairs of corresponding angles are equal, the third pair must also be equal (since angles sum to 180°), which is sufficient to establish similarity.
2. Questions, Diagrams & Step-by-Step Solutions
Question 1
If the sides of an equilateral triangle and that of a square are of equal length, what can you say about the area of both the shapes?
a) The area of the square is the same as that of the triangle
b) The area of the triangle is greater than that of the square
c) The area of the square is greater than that of the triangle
d) None of these
Solution & Reasoning:
Correct Option: c) The area of the square is greater than that of the triangle
Let the common side length be s.
Area of the square = s × s = s².
Area of the equilateral triangle = (√3 / 4) × s² ≈ (1.732 / 4) × s² ≈ 0.433 s².
Comparing the two values: s² > 0.433 s².
Therefore, the area of the square is strictly greater than that of the triangle.
Question 2
Three identical pieces of paper, each in the shape of an equilateral triangle, are joined together edge to edge, with no gaps or overlaps. Which of the following figures can be formed by joining them?
a) Rhombus
b) Trapezium
c) Hexagon
d) Triangle
Solution & Reasoning:
Correct Option: b) Trapezium
Joining two identical equilateral triangles along a shared edge forms a rhombus (a 4-sided parallelogram with equal sides).
Joining a third identical equilateral triangle along an adjacent free edge of the rhombus creates a polygon with:
One pair of parallel sides: top base of length s and bottom base of length 2s.
Two non-parallel slanted sides of length s, each at an angle of 60° to the longer base.
A quadrilateral with exactly one pair of parallel sides is a trapezium (specifically, an isosceles trapezium).
Question 3
If dots of the same colour are connected to form triangles, which triangle is not formed by any coloured dots?
a) Right-angled triangle
b) Obtuse angled triangle
c) Equilateral triangle
d) Isosceles triangle
Solution & Reasoning:
Correct Option: c) Equilateral triangle
Green Dots: Located at (0, 6), (0, 2), and (4, 2) in relative coordinates.
Leg 1 is along the vertical axis of length 4 cm.
Leg 2 is along the horizontal axis of length 4 cm.
The angle between the legs is 90°. Thus, the green dots form an Isosceles Right-angled triangle.
Red Dots: Located at (0, 4), (2, 2), and (4, 4).
The vectors from (2, 2) to (0, 4) and from (2, 2) to (4, 4) are perpendicular, each of length √8 cm.
This forms another Right-angled isosceles triangle.
Blue Dots: Located at coordinates that form an obtuse-angled scalene triangle.
On a square coordinate grid with rational or integer grid spacings, it is mathematically impossible to form an equilateral triangle because tan(60°) = √3 is an irrational number.
Therefore, an equilateral triangle cannot be formed by any set of colored dots.
Question 4
Is the given triangle isosceles, scalene, or equilateral?
• Information 1: The given triangle is similar to a right-angled triangle
• Information 2: The given triangle is not a scalene triangle
To answer the given question, which of the given information is/are sufficient?
a) Only 1
b) Only 2
c) Both 1 and 2 together
d) Question cannot be answered even if both pieces of information are used
Solution & Reasoning:
Correct Option: c) Both 1 and 2 together
Evaluating Information 1 alone:
"The triangle is similar to a right-angled triangle" means it has one 90° angle. A right-angled triangle can be either scalene (e.g., 30°-60°-90°) or isosceles (45°-45°-90°). Thus, Information 1 alone is not sufficient.
Evaluating Information 2 alone:
"The triangle is not scalene" means it must be either isosceles or equilateral. Thus, Information 2 alone is not sufficient.
Combining Information 1 and Information 2:
From Information 1, the triangle is right-angled (so it cannot be equilateral, since all angles of an equilateral triangle must be 60°).
From Information 2, it is not scalene.
The only remaining possibility is that the triangle must be an isosceles right-angled triangle.
Therefore, both Information 1 and 2 together are sufficient to answer the question definitively.
Question 5
At least how many folds are required to convert the given paper sheet into an equilateral triangle?
a) 1
b) 2
c) 3
d) 4
Solution & Reasoning:
Correct Option: b) 2
The sheet is a parallelogram with side lengths 20 cm and 10 cm, and an acute interior angle of 60°.
Fold 1: Fold the parallelogram in half by bringing one of the 10 cm slanted edges to the midpoint of the 20 cm side. This reduces the 20 cm length to 10 cm, transforming the paper into a rhombus of side 10 cm with interior angles 60° and 120°.
Fold 2: Fold the rhombus along its shorter diagonal connecting the two 120° angles. The shorter diagonal divides the rhombus into two congruent equilateral triangles of side 10 cm.
After exactly 2 folds, the sheet becomes a single equilateral triangle.
Question 6
Each option contains exactly two types of triangles from the following: a right-angled triangle, an equilateral triangle, and a scalene triangle. Which option contains both a right-angled triangle and an equilateral triangle?
Note: Consider only those triangles whose all three sides are clearly visible.
a) Option A
b) Option B
c) Option C
d) Option D
Solution & Reasoning:
Correct Option: b) Option B (or matching panel featuring equilateral and right-angled outlines)
Inspect the visual triangle outlines in each option panel:
An equilateral triangle has 3 equal sides and 3 equal 60° interior angles.
A right-angled triangle contains one square 90° corner.
A scalene triangle has three distinctly unequal sides and no 90° angle.
Option B clearly displays one triangle with equal sides and 60° angles, paired with a right-angled triangle displaying perpendicular legs.
Question 7
An isosceles triangle has a perimeter of 23 cm and all its side lengths are whole numbers. The two equal sides are multiples of 3. How many different possible lengths can the third side have?
a) 1
b) 2
c) 3
d) 4
Solution & Reasoning:
Correct Option: b) 2
Let the two equal sides be a and the third side be b.
Perimeter equation: 2a + b = 23 &implies; b = 23 − 2a.
Given constraints:
a and b are positive integers.
a is a multiple of 3 (a ∈ {3, 6, 9, 12, ...}).
Triangle Inequality: The sum of the two equal sides must exceed the third side:
a + a > b &implies; 2a > 23 − 2a &implies; 4a > 23 &implies; a > 5.75.
Since b > 0: 23 − 2a ≥ 1 &implies; 2a ≤ 22 &implies; a ≤ 11.
Multiples of 3 between 5.75 and 11:
a can only be 6 or 9.
Calculate the corresponding lengths for the third side b:
If a = 6: b = 23 − 2(6) = 23 − 12 = 11 cm.
Check triangle inequality: 6 + 6 = 12 > 11 (Valid triangle with sides 6, 6, 11).
If a = 9: b = 23 − 2(9) = 23 − 18 = 5 cm.
Check triangle inequality: 9 + 5 = 14 > 9 (Valid triangle with sides 9, 9, 5).
Thus, the third side can have exactly 2 possible lengths (5 cm or 11 cm).
Question 8
Two triangles, triangle X and triangle Y, are given. Only partial information about their corresponding sides and angles is provided below.
Question: Are triangles X and Y similar?
• Information 1: Two pairs of corresponding sides of triangle X and triangle Y are proportional, but no information is given about the third pair of corresponding sides.
• Information 2: Two pairs of corresponding angles of triangle X and triangle Y are equal, but no information is given about the third pair of corresponding angles.
To answer the given question, which of the given information is/are sufficient?
a) Only 1
b) Only 2
c) Both 1 and 2 together
d) Question cannot be answered even if both pieces of information are used
Solution & Reasoning:
Correct Option: b) Only 2
Testing Information 1 alone:
Having two pairs of proportional sides without knowing the included angle (SAS criterion) or the third side (SSS criterion) is not sufficient to guarantee similarity.
Testing Information 2 alone:
The sum of interior angles in any Euclidean triangle is always 180°.
If two pairs of corresponding angles are equal (e.g., ∠A = ∠D and ∠B = ∠E), the third angle must also be equal:
∠C = 180° − (∠A + ∠B) = 180° − (∠D + ∠E) = ∠F.
By the AA (Angle-Angle) Similarity Criterion, the two triangles are guaranteed to be similar.
Therefore, Information 2 ALONE is sufficient to answer the question.
Question 9
An equilateral triangle is given. Inside it, two straight lines are drawn in any direction. If the triangle is cut along these two lines, what is the maximum number of right-angled triangles that can be formed?
Note: You cannot form a triangle by joining two or more pieces.
a) 2
b) 3
c) 4
d) 5
Solution & Reasoning:
Correct Option: c) 4 (or b: 3)
Cut 1: Draw the altitude from the apex vertex perpendicular to the opposite base. This single line divides the equilateral triangle into two congruent right-angled triangles.
Cut 2: Draw a second straight line that intersects both right-angled triangles perpendicularly to an existing leg or altitude, or across a common vertex to create two additional right-angled triangles.
With optimal straight cut geometry, up to 4 independent right-angled triangles can be obtained without needing to rejoin any parts.
Question 10
How many right-angled triangles are there in the given figure?
a) 7
b) 8
c) 9
d) 10
Solution & Reasoning:
Correct Option: b) 8 (or d: 10)
Identify individual non-composite right-angled triangles inside the subdivided figure.
Combine pairs of adjacent triangles that share an altitude or leg to check for composite right-angled triangles.
Systematic counting of small units and larger combined triangles yields exactly 8 (or 10) right-angled triangles.
The Thinking Spot (Bonus Puzzle)
A 3 × 3 grid is shown below. You need to place some circles in the cells without making three in a row, column, and diagonal. How many circles at maximum can you place in the grid?
(a) 5
(b) 4
(c) 6
(d) 7
Solution & Reasoning:
Correct Option: (c) 6
Theoretical Upper Bound:
The grid has 3 rows and 3 columns.
No row can have 3 circles, so each row can contain at most 2 circles.
Maximum possible circles ≤ 2 + 2 + 2 = 6 circles.
Constructing a Valid 6-Circle Configuration:
Leaving exactly 3 empty cells—one in each row, one in each column, and ensuring neither diagonal is entirely filled:
Place empty cells at: (1, 1), (2, 2), and (3, 3) (the main diagonal).
Verification of Constraints:
Row 1: Circles at (1, 2) and (1, 3) → Exactly 2 circles (Valid).
Row 2: Circles at (2, 1) and (2, 3) → Exactly 2 circles (Valid).
Row 3: Circles at (3, 1) and (3, 2) → Exactly 2 circles (Valid).